BEE Control Systems Theory 5 — Questions and Answers
Question 1: The controllability matrix for a system with matrices A and B is formed as [B, AB, A²B, ...]. A system is controllable if this matrix has:
- All eigenvalues in the left-half plane
- Rank equal to the order n of the system (Correct answer)
- A symmetric positive-definite form
- Determinant equal to zero
Correct answer: Rank equal to the order n of the system
A system is controllable (Kalman's criterion) if and only if the controllability matrix [B AB A²B ... Aⁿ⁻¹B] has full rank n.
Question 2: In a discrete-time control system with sampling period T, the z-transform variable z and the Laplace variable s are related by:
- z = e^(sT) (Correct answer)
- z = sT
- z = 1/(sT)
- z = ln(sT)
Correct answer: z = e^(sT)
The mapping between continuous and discrete domains is z = e^(sT), which maps the left-half s-plane to the interior of the unit circle in the z-plane.
Question 3: A system exhibiting limit cycle behavior is best classified as:
- Linear and time-invariant
- Nonlinear, with a sustained oscillation at a fixed amplitude and frequency (Correct answer)
- Marginally stable and linear
- Unstable with exponentially growing oscillations
Correct answer: Nonlinear, with a sustained oscillation at a fixed amplitude and frequency
Limit cycles are self-sustained oscillations characteristic of nonlinear systems, with fixed amplitude and frequency independent of initial conditions within a region.
Question 4: Which of the following actions does a lead compensator primarily perform on the Bode plot?
- Adds phase lag at high frequencies to reduce noise sensitivity
- Adds phase lead near the crossover frequency to improve phase margin (Correct answer)
- Increases low-frequency gain to reduce steady-state error
- Reduces the gain margin by shifting the phase crossover frequency
Correct answer: Adds phase lead near the crossover frequency to improve phase margin
A lead compensator contributes positive phase (phase lead) near the crossover frequency, improving phase margin and transient response speed.
Question 5: For an underdamped second-order system, the percent overshoot depends only on:
- The natural frequency ωn
- The damping ratio ζ (Correct answer)
- The closed-loop gain
- The input signal amplitude
Correct answer: The damping ratio ζ
Percent overshoot = exp(−πζ/√(1−ζ²)) × 100%, which depends solely on the damping ratio ζ.
Question 6: The sensitivity function S(s) = 1/[1 + G(s)H(s)] in a feedback control system quantifies:
- How much the closed-loop transfer function changes relative to changes in the plant (Correct answer)
- The ratio of output to disturbance at the plant input
- The open-loop frequency response
- The steady-state gain of the controller
Correct answer: How much the closed-loop transfer function changes relative to changes in the plant
The sensitivity function S(s) measures the relative change in closed-loop transfer function due to relative changes in the plant G(s), with small |S| indicating robustness.
Question 7: In the Ziegler-Nichols ultimate gain tuning method, the controller is first set to proportional-only mode and the gain is increased until:
- The steady-state error reaches zero
- The system output exhibits sustained oscillations (Correct answer)
- The rise time is minimized
- The overshoot exceeds 50%
Correct answer: The system output exhibits sustained oscillations
In the Ziegler-Nichols method, the ultimate gain Ku is found by increasing K until the system oscillates at constant amplitude (marginally stable), then the ultimate period Tu is recorded.
BEE Control Systems Theory 5
The controllability matrix for a system with matrices A and B is formed as [B, AB, A²B, ...].
A system is controllable if this matrix has: